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A sequence \((u_n)\) is defined for \(n \ge 0\) by the integral:$$ u_n = \int_0^1 \frac{x^n}{1+x} \,dx $$
  1. Calculate \(u_0\).
  2. Prove that for any integer \(n \ge 0\), the recurrence relation \(u_{n+1} + u_n = \frac{1}{n+1}\) holds.
  3. Hence, deduce the value of \(u_1\).

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